Fill remaining vertical space - only CSS

The modern way to achieve this is using CSS Flexbox (see the MDN Web Docs).

html, body {
  height: 100%;
}

.wrapper {
  display: flex;
  flex-direction: column;
  width: 300px;
  height: 100%;
}

.first {
  height: 50px;
}

.second {
  flex-grow: 1;
}

<div class="wrapper">
  <div class="first" style="background:#b2efd8">First</div>
  <div class="second" style="background:#80c7cd">Second</div>
</div>

You can use CSS Flexbox instead another display value, The Flexbox Layout (Flexible Box) module aims at providing a more efficient way to lay out, align and distribute space among items in a container, even when their size is unknown and/or dynamic.

Example

/* CONTAINER */
#wrapper
{
   width:300px;
   height:300px;
    display: -webkit-box; /* OLD - iOS 6-, Safari 3.1-6 */
    display: -moz-box; /* OLD - Firefox 19- (buggy but mostly works) */
    display: -ms-flexbox; /* TWEENER - IE 10 */
    display: -webkit-flex; /* NEW - Chrome */
    display: flex; /* NEW, Spec - Opera 12.1, Firefox 20+ */
    -ms-flex-direction: column;
    -moz-flex-direction: column;
    -webkit-flex-direction: column;
    flex-direction: column;
}

/* SOME ITEM CHILD ELEMENTS */
#first
{
   width:300px;
    height: 200px;
   background-color:#F5DEB3;

}

#second
{
   width:300px;
   background-color: #9ACD32;
    -webkit-box-flex: 1; /* OLD - iOS 6-, Safari 3.1-6 */
    -moz-box-flex: 1; /* OLD - Firefox 19- */
    -webkit-flex: 1; /* Chrome */
    -ms-flex: 1; /* IE 10 */
    flex: 1; /* NEW, */
}

jsfiddle Example

If you want to have full support for old browsers like IE9 or below, you will have to use a polyfills like flexy, this polyfill enable support for Flexbox model but only for 2012 spec of flexbox model.

Recently I found another polyfill to help you with Internet Explorer 8 & 9 or any older browser that not have support for flexbox model, I still have not tried it but I leave the link here

You can find a usefull and complete Guide to Flexbox model by Chris Coyer here

You can do this with position:absolute; on the #second div like this :

FIDDLE

CSS :

#wrapper{
    position:relative;
}

#second {
    position:absolute;
    top:200px;
    bottom:0;
    left:0;
    width:300px;
    background-color:#9ACD32;
}

EDIT : Alternative solution

Depending on your layout and the content you have in those divs, you could make it much more simple and with less markup like this :

FIDDLE

HTML :

<div id="wrapper">
    <div id="first"></div>
</div>

CSS :

#wrapper {
    height:100%;
    width:300px;
    background-color:#9ACD32;
}
#first {
    background-color:#F5DEB3;
    height: 200px;
}